This article is devoted to the effects of constant vorticity in the setting of weakly nonlinear surface gravity waves
for which modulation theory is a classical tool.
Under consideration is the asymptotic scaling regime where approximate solutions are constructed as
slow modulations of monochromatic waves in two space dimensions.
For this problem, [1]} derived a cubic nonlinear Schrödinger (NLS) equation governing the envelope of surface gravity waves
on finite depth using the method of multiple scales. Their analysis was extended to gravity-capillary waves by [2]}
and to hydroelastic waves by [3]}.
Similarly, [4]} carried out the perturbation calculations up to an order higher and obtained a Dysthe equation
for gravity-capillary waves with constant vorticity on infinite depth. We also point out the earlier work of [5]} who proposed an NLS equation for gravity waves in the presence of linear shear confined to a finite-depth layer near the free surface.
